Theorem neisspw 12942

Description: The neighborhoods of any set are subsets of the base set. (Contributed by Stefan O'Rear, 6-Aug-2015.)

Hypothesis

Ref	Expression
neifval.1	|- X = U. J

Assertion

Ref	Expression
neisspw	|- ( J e. Top -> ( ( nei ` J ) ` S ) C_ ~P X )

Proof of Theorem neisspw

Dummy variable v is distinct from all other variables.

Step	Hyp	Ref	Expression
1		neifval.1	. . . . 5 |- X = U. J
2	1	neii1 12941	. . . 4 |- ( ( J e. Top /\ v e. ( ( nei ` J ) ` S ) ) -> v C_ X )
3		velpw 3573	. . . 4 |- ( v e. ~P X <-> v C_ X )
4	2, 3	sylibr 133	. . 3 |- ( ( J e. Top /\ v e. ( ( nei ` J ) ` S ) ) -> v e. ~P X )
5	4	ex 114	. 2 |- ( J e. Top -> ( v e. ( ( nei ` J ) ` S ) -> v e. ~P X ) )
6	5	ssrdv 3153	1 |- ( J e. Top -> ( ( nei ` J ) ` S ) C_ ~P X )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   e. wcel 2141   C_ wss 3121   ~Pcpw 3566   U.cuni 3796   ` cfv 5198   Topctop 12789   neicnei 12932

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-top 12790  df-nei 12933

This theorem is referenced by: (None)